 One of the key activities of mathematics is generalizing basic concepts. So if I take some vector in Rn, then the dot product of that vector with itself can be interpreted as the square of the Euclidean length of the vector. The norm is a generalization of this concept. So as I have some vector space, a norm is a function that goes from the vector space into the real numbers with the following properties for all vectors in our vector space and all scalars in our real numbers. First, while the norm is going to be a real number, the norm of a vector will be 0 if and only if the vector itself is the 0 vector. For all other vectors, the norm of the vector must be greater than 0. Next, we have our scalar multiple property. The norm of the scalar multiple of a vector is the absolute value of the scalar times the norm of the vector itself. Finally, we have the triangle inequality. The norm of a sum of two vectors must be less than or equal to the sum of the norms of the vectors. Just a quick note, we can actually generalize this concept even further if we substitute any scalar field for the real numbers. The concept of a norm is the generalization of the concept of Euclidean distance. So let's see how that plays out. Another way we can generalize this concept is to use what's called the p-norm. So let's say I have some vector in Rn. I'll define the p-norm as the p-th root of the sum of the p-th powers of the absolute value of the components of the vector. And because mathematicians like taking things too extreme, we'll also talk about the infinity norm, which is going to be the maximum of the absolute values of the vector components. Now, because this defines p-norms in a very general fashion, p could be anything. So we could have seven norms or even fractional norms, like a three-fifths norm. The most commonly used norms are the one norm, the two norm, and the infinity norm. For example, let's take a vector in R5 and find the two norm, the one norm, and the infinity norm. So remember that in general the p-norm is the p-th root of the sum of the p-th powers of the absolute values of the vector components. So the two norm is going to be the one-half power of the sum of the tooth powers, the squares of the absolute value of the vector components. And so we'll find the sum of the squares of the absolute values and then take the square root to find our norm. Similarly, the one norm will be the one-th root of the sum of the absolute values of the vector components. Finally, the infinity norm will just be the largest of the absolute values of the vector components. At this point, it's worth retelling a riddle once asked by Abraham Lincoln. How many legs will a dog have if you call a tail a leg? The answer? Four, because calling a tail a leg doesn't make it a leg. So except for the opportunity to tell a historically important riddle, why would I bring this one up? The important thing is that even though we call p-norms norms, that doesn't make them norms, we actually have to prove they are norms. So suppose I want to prove that the one norm is in fact a norm. We'll go back to our requirements and we'll think about these as a checklist. First, the one norm is zero if and only if the vector is the zero vector. Next, the one norm has to be greater than or equal to zero for all vectors. Next, we have our scalar multiplication requirement that the norm of the scalar multiple of a vector is the absolute value of the scalar times the norm of the vector. And finally, there is the triangle inequality, the norm of the sum of two vectors must be less than or equal to the sum of the norms of the vectors. So these are the things that must be true in order for something to be a norm and so we'll check to make sure that our one norm meets these requirements. First, we'll go ahead and set down our definition of what the one norm is and then we might make a few observations. First, the absolute value of the v-i's has to be greater than or equal to zero, which means that this sum is going to be greater than or equal to zero and so the one norm for any vector will be greater than or equal to zero. So our second requirement is met. Next, none of the summons can be negative, so in order for the sum to be zero, they all have to be zero. And so the only way for the one norm to be zero is going to be when all of the components of the vector are zero and so our one norm is zero only for the zero vector. So our first requirement is also met. Next, we'll check to see how the norm acts under scalar multiplication. So I want to find the one norm of a scalar multiple of my vector. So applying our definition, that's going to be the sum of the absolute values of the scalar multiples of the vector components and because c and the vector components are real numbers, I know that absolute value can be split and I can factor out the absolute value of c and this sum inside the parentheses is just going to be the one norm of the vector and so that means that my norm of the scalar multiple of my vector is in fact the absolute value of the scalar multiple times the norm of the vector itself and so our third requirement is also met. Finally, we want to check our triangle inequality so we need to find the one norm of the sum of two vectors. So again, we can find that one norm by finding the sum of the absolute value of the sum of the vector components and again, because our vector components are real numbers, we know that the absolute value of a sum is guaranteed to be less than or equal to the sum of the absolute values. Now we can separate our absolute values. Here, we have the absolute values of the components of the first vector and here we have the absolute values of the components of the second vector so we can rewrite our sum and this first sum is the same as the one norm of the first vector and the second part is the same as the one norm of the second vector and so our triangle inequality holds for the one norm. So our one norm meets all of the required properties of a norm and so we're able to say that the one norm is in fact a norm.